In fact, while Doi [1]}, Murata [2]} show that smoothing
estimates and the usual decay estimates do not hold in the presence of
trapping, Ikawa [3]} shows that one still obtains local decay
estimates for the Laplacian dynamics on \({\mathbb {R}}^n\)  with several convex obstacles
removed. In the meantime, the microlocal analysis on manifolds with unstable closed geodesics, of the resolvent of the Laplacian on the one hand, and the
Schrödinger evolution on the other hand, has grown into a vast area in and of itself which
is intimately connected to the semiclassical analysis of scattering resonances. See for example the
recent research monograph [4]} on Resonances for homoclinic trapped sets, or Dyatlov's introduction to
the fractal uncertainty principle [5]}.
